Obsidian/Zotero/001_artiklid/Road Safety Indicators/cenajroadaccidentfatalitiesforecasting2024.md

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category: literaturenote citekey: cenajroadaccidentfatalitiesforecasting2024 title: "Road Accident Fatalities Forecasting Models using Smeed’s Regression Analysis: A Case Study" authors: "Cenaj, Erjola; Dervishi, Raimonda" year: 2024 date: 2024-00-00 2024 doi: 10.24018/ejeng.2024.9.6.3213 publication: European Journal of Engineering and Technology Research url: "https://doi.org/10.24018/ejeng.2024.9.6.3213" zotero_key: 2SGNNE3S zotero_storage: VEQ5AUQ2 collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Cenaj, Erjola"

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RESEARCH ARTICLE

Road Accident Fatalities Forecasting Models using Smeed's Regression Analysis: A Case Study

Erjola Cenaj\* and Raimonda Dervishi

ABSTRACT

Fatalities and injuries triggered by road traffic accidents are worthwhile traffic safety problems, especially for developing and developed countries. Several academic works have convinced that using data from a given group, a weighty relationship between the number of road traffic fatalities (RTFs) and registered vehicles and population size is designed. This paper uses historical data from over ten years in Albania to identify the relationship between the same factors. The study divulges that such a relationship cannot be impacted with a good degree of exactness by Smeed's Equation (SE). Additionally, road accident prediction models by the structural form of Smeed's regression method enable estimating RTFs tied to population size and number of active vehicles. The modified forms of SE are the key to a satisfactory match in predicting RTFs in Albania.

Keywords: Number of vehicles, regression analysis, Smeed's equation, traffic accident fatalities.

Submitted: October 10, 2024

Published: November 28, 2024

10.24018/ejeng.2024.9.6.3213

Department of Mathematical Engineering, Faculty of Mathematical and Physical Engineering, Polytechnic University of Tirana, Albania.

*Corresponding Author: e-mail: erjola\_cenaj@yahoo.com

1. Introduction

Fatalities and injuries due to road traffic accidents are substantial traffic safety problems for both developing and well-constructed countries. According to the statistics of the World Health Organization (WHO), worldwide, 1.3 million people die, and as many as 20–50 million people are also injured annually linked with traffic accidents. The same statistics point out that approximately 90% of fatalities from road traffic accidents transpire in low-andmiddle-income countries, despite the fact they hold a 50% share of motor vehicle ownership in the world [1].

On the other hand, road traffic accidents are the leading cause of all injuries and are representative of most deaths, crafting road traffic injuries the 9th leading matter of disability-adjusted life years lost worldwide [2]. Impetuosity and insanity, in many cases, are the foremost sources, especially in Balkan countries, and lack of proper sense of using traffic signs are some of RTFs sources. However, it is effective, entitizing other components such as population size, the number of vehicles in use, the total length of roads, and the population density, as if these factors can be translocated to reduce the future number of accidents [3], [4].

Many studies have shown that population size and number of registered vehicles are dominant factors affecting road traffic fatalities worldwide [5], [6].

Smeed's Equation (SE) is the earliest model used to approximate the number of fatalities in road traffic accidents, comprising the independent variables of population and number of registered vehicles and the dependent variable of the number of fatalities [7]. It has been found that SE is adequate for utilization in some developed countries [8], and according to Ponnaluri [9], it is a poorly method in developing countries [10]–[12].

The process of RTFs evaluation was introduced for the first time in 1949, by Smeed. Smeed started to work based on available data linked to road traffic fatalities (RTFs), official data on active vehicle numbers, and population size referring to 1938 from the group of 20 countries selected (mostly EU). Then, he discovered a nonlinear relationship among RTFs and influencing demographic and motorization rates through the mathematical expression given in (1) [13]:

$$\frac{F}{V} = 0.0003 \left(\frac{V}{P}\right)^{\frac{-2}{3}} \tag{1}$$

where

F – number of road traffic fatalities (RTFs)

V – number of registered vehicles

P – population size

The Smeed equation provides a good match referring to available data from the selected countries for that year. The factors of experiencing possible road traffic accidents are coupled to RTFs, population size, and the number of registered vehicles. This approach is used to carry out important sizes handling identical factors as Smeed. We used the number of RTFs, official vehicle number registered vehicles, and population size, including the time fraction 2014–2023. As in many other countries worldwide, the population size and number of registered vehicles are dominant factors affecting road traffic fatalities in Albania. The study shows that Smeed's Equation is inadequate to be used for definitively predicting RTFs in Albania over the study period. The model is tested using the statistics from INSTAT Albania. It has been shown that these modified forms of SE lead to a satisfactory fit for the tested period.

2. Data Source

In this research work, the factors mentioned in the Smeed equation, such as population size, number of registered vehicles, and number of road traffic fatalities for the tested period under consideration, 2014–2023, are provided from the domestic official sources, Albanian Statistical Institute (INSTAT) [14].

According to INSTAT, around 16,192 road traffic accidents in Albania have been declared during the last 10 years. Furthermore, around 24,020 people were involved in road accidents, split into 2199 cases of fatalities, and 19,622 were injured between the 2014–2023 period, as depicted in Table I.

3. Methodology

3.1. Application of Smeed's Regression Model to Estimate RTFs

Smeed's model is an empirical rule that binds RTFs to variables, the vehicle registered, and the population size for a given region, or country. As it is expected, increasing the number of vehicles will decrease fatalities per vehicle. This principle was first introduced based on 1938 work, which was employed later to test RTFs in 1960 up to1967 period or even later in many other scientific efforts included in the research work of [15].

TABLE I: Statistical Data for the Tested Period (2014–2023)

Year	Fatalities	Population	No. of vehicle
2014	264	2,895,947	415,121
2015	270	2,893,005	443,227
2016	269	2,875,592	479,217
2017	222	2,876,591	460,299
2018	213	2,870,324	500,894
2019	227	2,862,427	499,779
2020	181	2,845,955	539,497
2021	197	2,829,741	593,280
2022	164	2,793,592	639,379
2023	192	2,761,785	699,337

Smeed's equation model can be given from (2) [13]:

$$\frac{F}{V} = \alpha \cdot \left{ \left( \frac{V}{P} \right)^{\beta} \right} \cdot \varepsilon \tag{2}$$

where

F – number of RTFs

V – number of registered vehicles

P – population size

E – error term

α and β – parameters that are a matter of estimation

On the other hand, (2) can be expressed as following in (3):

$$Y = aX^{\beta}\varepsilon\tag{3}$$

where

X – the predictor variable is V/P (vehicle density)

Y the dependent variable, F/V (fatality rate per vehicle)

In the process of evaluating RTFs, Smeed was based on the least squares regression (LSR) techniques to assess the parameters linked to (3). However, the LSR approach does not allow parameter variability and it is very sensitive to infraction of the normality assumption.

3.2. Literature Review on the Application of Modified Smeed's Regression Model

A lot of efforts are undertaken by many authors in the field of RTFs theory. The research work and contribution of Koren and Borsos have led to a modified form of the original Smeed's equation as can be given in the following mathematical expression given in (4) [16]:

$$\frac{F}{P} = \alpha \left(\frac{V}{P}\right)^{\beta} \varepsilon \tag{4}$$

where α and β are the parameters necessary to be determined, for the known variables, such as the number of road traffic fatalities (F), the number of registered vehicles (V), and the population (P).

After transforming (4) and supposing both side logarithms, to base e, the nonlinear model in (4) can be expressed in a linear form as follows in (5):

$$Y_i = \alpha_0 + \beta X + e_i \tag{5}$$

where

Yi – ln(F/P), Xi = ln(V/P)

α⁰ – lnα

ei – lnε

i – 1, 2, ... , n

Moreover, during this transformation, it is required that ei, i = 1, 2 ... , n are iid distributed N(0, σ2).

Further on, using least squares regression (LSR) method to estimate the mentioned above parameters, should pass through employing expression in (6):

$$\hat{\beta} = \frac{\sum_{i=1}^{n} x_i yi - \frac{1}{n} \sum{i=1}^{n} xi \sum{i=1}^{n} yi}{\sum{i=1}^{n} xi^2 - \frac{1}{n} (\sum{i=1}^{n} x_i)^2}$$ (6)

Or *α*ˆ ⁰ is linked as following in (7):

$$\hat{\alpha}_0 = \overline{y} - \hat{\beta}\overline{x} \tag{7}$$

Using the equation α⁰ = lnα, then the least squares estimation of the parameter α can be evaluated using (8):

$$\hat{\alpha} = e^{\hat{\alpha}_0} \tag{8}$$

Thus, in terms of initial variables, the expression given in (9) is carried out:

$$\frac{F}{P} = \hat{\alpha} \left(\frac{V}{P}\right)^{\hat{\beta}} \tag{9}$$

3.3. Test of Regression

The regression hypothesis that none of the x's predict y can be expressed as H0: β = 0. The modeling process does not include β = 0 as part of our hypothesis, enabling the restriction of y to ensure realizing an intercept of zero.

We can test H0: β = 0 by (10):

$$F = \frac{SSR}{SSE/(n-2)} \tag{10}$$

It is simply distributed as F (*1, n*−2) when H0: β¹ = 0 is true. If F > F (*α; 1, n*−2), then it is worth rejecting H0 from simulations.

4. Discussion and Results

4.1. Estimation of RTFs using Smeed's Regression Model in Albania

The results of our research work are introduced in Table II.

As it can be depicted from the results (Table II), the application of Smeed's equation leads to an overestimation of the number of RTFs in Albania.

The result shows that on average, the expected fatalities estimated by Smeed's formula exceed the observed values by 4.8%, which compared to official data's (2.19%) is higher. Given the relatively large deviations between observed and expected values of fatalities as stated in Table II, Smeed's equation is ineffective in predicting RTFs in Albania.

Using (2) and performing LSR technique on the data given in Table I, we get the mathematical expression as given in (11):

$$\frac{F}{V} = 0.0000232 \left(\frac{V}{P}\right)^{-1.6992} \tag{11}$$

TABLE II: Simulation Results on Estimated Fatalities based on Smeed's Regression Models

Year	Fatalities	F ˆ	Error (%)	F ˜	Error
2014	264	454	190 (71.9%)	261	3 (1.2%)
2015	270	464	194 (71.8%)	249	21 (7.7%)
2016	269	474	205 (76.2%)	233	36 (13.4%)
2017	222	468	246 (110%)	240	−18 (8.1%)
2018	213	481	268 (125%)	226	−13 (6.1%)
2019	227	479	252 (111%)	225	2 (0.8%)
2020	181	490	309 (170%)	211	−30 (16.5%)
2021	197	504	307 (155%)	196	1 (0.5%)
2022	164	512	348 (212%)	182	−18 (10.9%)
2023	192	560	368 (191%)	167	26 (13.5%)
Total	2199	4886		2190

Note: F = number of RTFs, F (as in the 5th column title) = estimate of F from Smeed's formula.

TABLE III: ANOVA Test: The Estimated Smeed's Regression Model ANOVA df SS MS F Significance F Regression 1 0.859457 0.859457 76.82324 2.25E-05 Residual 8 0.0895 0.011187 Total 9 0.948956

TABLE IV: ANOVA Test: The Estimated Modified Smeed's

Regression Model
ANOVA	df	SS	MS	F	Significance F
Regression	1	0.145557	0.145557	13.01249	0.006909
Residual	8	0.089488	0.011186
Total	9	0.235045

Using (11), the result of this application, the estimated *F*˜ , which portrays the estimate of F from (11) is introduced in Table II.

From the simulation results it is shown that the application of Smeed's regression model is adequate for assessing the RTFs in Albania (Table II). The result shows that on average, the expected fatalities as estimated by Smeed's formula exceeded the observed by 2.19% but results tightly to the average of real fatalities.

Table III presents the analysis of variance from the estimated Smeed's regression model. From the test of regression, the statistic:

$$F = \frac{regression \, mean \, square}{residual \, mean \, square}$$

where H0 is true for a significance level of 0.05, then it has Fisher distribution with 1 and n-2 degrees of freedom. If the calculated value of F is greater than F(0.05; 1, n-2), it necessitates the rejection of H0. Since Fcalculated = 76.82 > 5.32 = F(0.05;1,8), the test is significant at the 5% accuracy level. From the simulation results and analyses, it is concluded that a linear relationship between the expected value of y = ln(F/V) and x = ln(V/P) is observed.

4.2. Estimation of RTFs using Modified Smeed's Regression Model in Albania

Using (4) and LSR method based on Table I, we get the mathematical expression as follows in (12):

$$\frac{F}{P} = 0.0000231 \left(\frac{V}{P}\right)^{-0.6992} \tag{12}$$

TABLE V: Estimated Fatalities from the Modified Smeed's

| Regression Model | | | | | | |------------------|------------|--------|-------------|--|--| | Year | Fatalities | F
ˆ | Error | | | | 2014 | 264 | 260 | 4 (1.5%) | | | | 2015 | 270 | 248 | 22 (8.15) | | | | 2016 | 269 | 232 | 36 (13.4%) | | | | 2017 | 222 | 239 | −17 (7.6%) | | | | 2018 | 213 | 225 | −12 (5.6%) | | | | 2019 | 227 | 224 | 3 (1.3%) | | | | 2020 | 181 | 210 | −29 (16%) | | | | 2021 | 197 | 195 | 2 (1%) | | | | 2022 | 164 | 181 | −17 (10.4%) | | | | 2023 | 192 | 167 | 25 (13%) | | | | Total | 2199 | 2181 | | | |

	Coefficient		Standart error		R square
	Smeed's method	Modified Smeed's method	Smeed's method	Modified Smeed's method	Smeed's method	Modified Smeed's method
αˆ	−10.6722	−10.6724	0.331394	0.331372	0.90	0.62
βˆ	−1.69923	−0.69929	0.193868	0.193855

TABLE VI: Regression Results: Estimated Coefficients (*α*ˆ;*βˆ)*

Table IV presents the analysis of variance from the estimated modified Smeed's regression model. From the regression test, if the calculated value of F is greater than F(0.05; 1, n-2), it is recommended to reject H0. Since Fcalculated = 13.01 > 5.32 = F(0.05;1,8) the test is significant at the 5% level. We conclude that there is a linear relationship between the expected value of y = ln(F/P) and x = ln(V/P).

The estimated *F*ˆ , which represents the estimate of F evaluated from (12), is given in Table V. From the simulation results (Table V), it is shown that the application of modified Smeed's regression model is adequate for Table V. Estimated fatalities from the modified Smeed's regression model assessing RTFs in Albania case. The result shows that, on average, the expected fatalities, as estimated by Smeed's formula, exceeded the observed by 2.181% but results tightly to the average of real fatalities.

In Table VI, the coefficients estimated from both regression models, the standard errors for the LSM, and the R Square coefficient are given. From the simulation results included in Table VI, the estimated coefficient *α*ˆ, β , and standard error for both coefficients are almost equal. The R² coefficients of the determination chosen model results are 0.90 and 0.62, respectively. These values recommend that Smeed's regression model properly fit the data of the Albanian case referring to the 2014–2023 period.

5. Conclusions

Smeed's equation is a valuable tool to forecast fatalities linking to independent predictors, population size, and number of registered vehicles based on a non-linear mathematical expression form. Referring to official data within the period 2016–2023, Smeed's Equation cannot be specified as a good approximation due to its low degree of accuracy. Based on the approved index of safety (0.0003) in SE [8], and compared to other case studies, it is clearly stated that the number of fatalities in Albania is apparently away from the international average. Furthermore, in this research work, the structural form of Smeed's regression method to assess fatalities was employed for estimating RTFs in Albania. Once again, referring to official data within the period 2016–2023, Smeed's Equation can be specified as a good approximation due to its high degree of accuracy. Performing F tests, a high statistical significance (p < 0.001) for both developed models are observed. Moreover, the results carried out in this scientific work are a good starting point for predicting traffic safety conditions in the future.

The prediction of RTFs is a major issue for the safety of human life, which should be addressed more carefully together with the problems identified in studies [17]–[19] constitute the full dimensions of safety towards a reliable transport sector that meets the general notion of reliability in developing countries in 2030 and beyond.

In conclusion, the developed models led to the conclusion that increasing the number of registered vehicles may decrease fatalities.

Conflict of Interest

The authors declare that they do not have any conflict of interest.

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